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Central-place foraging in a patchy environment
J. theor. Biol. (1986) 123, 35-43
Central-place Foraging in a Patchy Environment RICHARD F. GREEN
Department of Mathematical Sciences, University of Minnesota, Duluth, Minnesota 55812, U.S.A. AND ADELINE TAYLOR NUNEZ
Forest Wildlife Project, Cloquet Forestry Center, 175 University Road, University of Minnesota, Cloquet, Minnesota 55720, U.S.A. (Received 12 June 1985, and in revised form 18 March 1986) Early models of central-place foraging treated animals that search for prey in identical, homogenous patches. If patches vary in quality, then optimal foraging requires strategies based on time spent in a patch, and not simply on the type or number of prey found. In particular, a forager that takes no more than one prey from a patch should leave a patch after searching unsuccessfully for a certain fixed time. When patches are more variable, the forager should stay a shorter time in each patch, and the resulting rate of delivering prey to the central place will be lower. This implies that aggregation should be favored by prey faced with a singleprey-loading predator.
1. Introduction Central-place foraging is a special case of foraging in which animals return with prey to a nest or larder, and carry a limited n u m b e r of prey on each return. Orians & Pearson's (1979) model of central-place foraging describes an animal foraging in h o m o g e n e o u s patches in which prey are found randomly at a known, constant rate. Orians & Pearson consider two different cases and treat a characteristic problem for each. (1) A single-prey loader takes one prey per visit to a patch. If prey vary in quality (the energy they yield to the forager, for example), then the forager should take the highest quality prey. The forager should be more selective when travel time t o - a n d - f r o m the nest is greater. (2) A multiple-prey loader may take several prey per visit to a patch. Such a forager must decide how many prey to take on each visit. The forager should take more prey when travel time to-and-from the nest is greater. Lessells & Stephens (1983) treat a special case o f Orians & Pearson's model for a single-prey loader, and state that it makes no sense for such an animal to leave a patch without finding a prey. This is true if patches are identical and homogeneous, as assumed by Orians & Pearson, but it is not true if patches vary in prey density. In this paper, we consider a model of a single-prey loader that forages in patches that vary in prey density. 35
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2. The Model
Imagine a central-place forager, perhaps a hawk or owl, which can carry only one prey item to the nest on each return, and which forages for prey in patches that vary in prey density. On each foraging excursion the animal leaves the nest, searches a patch for a time, and if no prey are found, moves on to another patch and searches it for a time. The animal moves from patch to patch until it captures a prey and returns to the nest. When the animal resumes foraging, it does not return to the patch where the last prey was captured. For simplicity, we assume that all prey are the same value. We assume that search within a patch is systematic, and that prey are encountered randomly, at rate r, which is proportional to the density of prey in the patch. The time to encounter a prey in a patch would have an exponential distribution with density
f(t)=rexp(-rt)
for t > 0
(I)
and the expected time until a capture would be ~z = 1/r. We assume that prey density varies from patch to patch, and that the encounter rate, r, is a random variable, having a gamma distribution with parameters ~ and/3. That is, r has probability density function
/3°
f(r)-F(a) r
e -~r
(2)
This mathematically convenient assumption implies that the number of prey encountered by a forager spending a fixed time in a patch has a negative binomial distribution (see Pieiou 1977, pp. 122-123). In our model, the average travel time between patches is t(1), and the average travel time back and forth from a patch to the nest is t(2), with t ( 2 ) > t(1). Our model has four parameters: a,/3, t(1) and t(2). Since patches vary in prey density, the longer a forager spends in a patch without finding a prey, the worse that patch is likely to be. The best strategy is to stay in each patch until a prey has been found, or until a fixed stopping time, t, is reached without finding a prey. Various choices of t could be made, but the best value, t*, maximizes the long-term average rate of obtaining prey (which we will refer to as the "capture rate"), given by
R = 1/(t(2)+ET),
(3)
where E T is the expected time spent foraging and traveling between patches per prey found. Notice that the value of t* depends on a,/3, and t(1), but not on t(2), the travel time to-and-from the nest, since the maximum capture rate is achieved by the rule that minimizes the expected time to find a prey while foraging. McNamara & Houston (1985) propose a model similar to ours for a slightly different problem. Considering a forager searching patches that contain exactly one prey, they assume that the time to find a prey is a random variable having an exponential distribution with mean v (corresponding to the difficulty of finding the prey), which is itself a random variable having a gamma distribution. The r in our model equals 1/v in McNamara & Houston's model. This rather subtle difference
CENTRAL-PLACE FORAGING
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in the models leads to a substantial difference in conclusions when we consider the effect of environmental variability. CALCULATION
OF
ET
For each choice of a, 13 and t(1) we can find the optimal stopping time, t*, which minimizes ET. For any a, 13 and t, the probability that a forager will fail to find a prey after searching a patch for time t is given by Q(t) =
[/3/(/3 + t)]%
(4)
If T is the time that a forager would have to wait to find a prey in a randomly chosen patch, the probability distribution of T is given by P(T < - t) = F ( t ) --- 1 - Q ( t )
(5)
and its probability density is given by f ( t) = F'( t) = ct/3~[/3 + t] -~-~.
(6)
That is, the random variable T has a Pareto distribution with location parameter/3 and shape parameter c~. If a prey is found before time t in a patch, the expected time spent in the patch before finding that prey is
for a ¢ 1, and
for a = 1. The average time spent foraging on each excursion is given by E T = E [ t I T <- t] + ( t ( 1 ) + t ) Q / ( 1 - Q )
(8)
where ( t ( 1 ) + t) denotes the time spent foraging and traveling between patches for each patch in which a prey is not found, and Q / ( 1 - Q ) is the average number o f patches searched unsuccessfully. 3. Results
Without loss of generality, we assume that all environments have the same average prey density per patch, (for example, a / / 3 = 1, and, therefore, that a =/3). Thus, in analyzing our model, we need to consider two parameters, a and t(1), where a is a measure o f patch variability, and t(1) is related to the density of patches in the environment. We compare environments that have the same average prey density, but differ in (a) the variability in prey density among patches, and (b) the travel time between patches. We look at three things: (1) how the capture rate achieved depends on the strategy used (the time a forager is willing to remain in a patch
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without finding a prey) for ditterent parameters, (2) how the optimal value t* depends on the parameters, and (3) how the optimal capture rate depends on the parameters. For any set of parameter values we can find the capture rate, R, achieved by a forager staying in a patch for time, t. In Fig. 1, R is plotted against t for four different sets of parameter values. The rate does depend on t, but is rather insensitive to t, especially when patches are less variable. This insensitivity of the rate achieved to the strategy used is seen in other stochastic foraging models as well (Green, 1984). Notice that the capture rate is highest when the travel time between patches is short and variability of prey density among patches is low. Capture rate is lowest when the travel time between patches is long and variability of prey density among patches is high. 0-5( Manylow-variabilityporches 04( cc
~ F e w
~ 0 3( r~ o
Iow-voriobihtypatches
~ o t c h e s
o~ Few h~gh-vorlobJhtypotches
020 / /
8 010 000
V
000
i
r
i
I
100 200 3 O0 400 Stoppingtime,t
i
500
FIG. l. Long-term average capture rate, R, plotted against stopping time, t (the time a forager is willing to remain in a patch without finding a prey), for four different sets of environmental conditions. Travel time between patches, t( 1) = 0.2 ( m a n y patches), or t(1 ) = 1 (few patches), while patch variability, a = 0.5 (high variability), or a = 2 (low variability). In each case, average patch quality, a / f l = 1, and travel time to-and-from the nest, t(2) = 1.
The reason that a forager should leave a patch if no prey are found by time t* is that as time spent in the patch without finding a prey increases, the chance of finding a prey there decreases. When a patch is entered, prey are found at rate a/fl, but after time, t, has been spent in the patch without finding prey, the rate decreases to a/(/3 + t). Since we assume that a =/3 this rate is a / ( a + t). If a forager increases the time it is willing to stay in a patch before finding a prey, it will increase the chance of finding a prey in a patch (that is, it will decrease Q), and thus decrease the chance of having to waste time travelling between patches. On the other hand, being willing to stay in a patch for a longer time without finding prey will increase the time wasted searching poor patches. When travel time between patches, t(1), is longer, travel is more costly and the optimal value, t*, will be higher. When patches are more variable (that is, a is smaller), a forager discovers more quickly if a patch is poor, and t* will be lower. Figure 2(a) plots the optimal patch time, t*, against or, for two different values of t(1), the travel time between patches. As a increases, patches are less variable and a forager should be willing to stay longer in each patch. In Fig. 2(b), the
CENTRAL-PLACE 4 O0
39
FORAGING 0 50
(o)
Many patches t(1)=O 2
~320
040
Few patches t (1) = 1 0 / 1 o
~24c
8 "8 16C E
,t J
- 020 0
ooo o'4o o18o ~'zo ~o
2oo
i "~
///
Few patches f(1}=t 0
///
Many patches t(1)=O 2
0 8( 00(
//
8
// //
ts
050
/ I
010
/
/
/
/
/
0 O0
ooo o'4o o'8o ~'2o ~'~o 20o
Cl(lncreose as patch varlobi~ily decreases)
(:l (~ncremsesas patch voflabthty decreases)
FIG. 2. (a) Optimal stopping time, t*, and (b) long-term average capture rate achieved by a forager using the optimal stopping time, plotted aginst a, a measure of patch variability (increasing a means decreasing variability). Travel time between patches, t(1) = 0.2 (many patches), or t(1) = 1 (few patches). In each case, average patch quality, all3 = 1, and travel time to-and-from the nest, t(2) = 1.
maximum possible capture rate (that is, the rate achieved by the value of t* shown in Fig. 2(a)) is plotted against a. Notice that capture rate is highest when patches are least variable. If, for a//3 = 1 and t ( 2 ) = 1, patches are homogeneous (t~ =oo), the forager should stay in each patch until a prey is found (t* = ~ ) , and the capture rate would be R =0.5. This is the case considered by Orians & Pearson (1979). 4. Discussion REALISM A N D G E N E R A L I T Y OF THE M O D E L
The model we have considered is a special case of foraging, but the assumption that makes it different from other models--that patches vary in prey density--is realistic biologically. One of the consequences of our assumption is that foragers should sometimes leave a patch without finding a prey. This is consistent with observations of animals feeding. The particular model of patch variability we have assumed implies that the number of prey per patch of fixed size should have a negative binomial distribution. This distribution seems to fit some data for insects well (Southwood, 1966), and the choices o f a = 0-5 and 2 which we have used in Fig. 1 are biologically reasonable. Our assumption that all prey are the same leads to the conclusion that the optimal strategy, which consists of choosing a value of t*, does not depend on travel time to-and-from the nest. This conclusion will also hold if there are different types of prey of different value to the forager, but the forager does not choose among them. If several types of prey are considered, then the optimal choice of prey may depend on travel time, but, contrary to the conclusion o f Orians & Pearson (1979), who assumed that patches are homogeneous, the choice o f prey may also depend on the length o f time a forager has spent in a patch. If patches are homogeneous, then Orians & Pearson are correct that the decision whether to take a prey of given quality or to continue searching until a better prey is found does not depend on how long the forager has searched the patch, since the forager has learned nothing
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from the length of his search. However, if patches vary in quality, then as more time is spent in a patch the forager should be willing to settle for poorer prey. If prey vary in value, V, to a forager, but not in the time necessary to capture and handle them, and if R is the highest possible capture rate, then a prey should be taken whenever found if V> Rt(2) (9) while a prey should never be taken if V< R[t(2) - t(1)].
(10)
Prey of intermediate value should be taken only if the forager has spent sufficient time in a patch without taking a better prey. The value of t(2), the travel time to-and-from the nest, is determined by how far the animal forages from the nest. Following Orians & Pearson (1979), we have treated t(2) as a constant. Of course, animals should prefer a short t(2), that is, they should forage near their nests. However, if animals forage in the vicinity of their nests, the area might become depleted of prey. Orians & Pearson (1979) suggest that the choice of where to forage involves a trade-off between the cost of travelling to distant patches and the cost of foraging in depleted nearby patches. Optimal foraging theory might contribute to an understanding of how animals use territories. There has been little work in this direction. Andersson (1978, 1981) has argued that efficient foraging would lead to a territory being used with intensity decreasing linearly away from the central place. The evidence for this idea has been disputed by Aronson & Givnish (1983), who suggest possible alternative patterns of territory use. These alternatives do not take advantage of the ideas of optimal foraging theory. We do not propose any solution to the problem of how animals use, or should use, their territories, but we do suggest that optimal foraging theory must be used, and that theory must take account of the prey distribution. For example, our results show that the optimal foraging strategy and the capture rate depend on prey distribution as well as average prey density. The choice of whether or not to search in partially depleted patches near the nest depends on the capture rate that can be achieved there, which, in turn, depends on the original prey distribution and the pattern of foraging which has depleted the patches. 5. Conclusions
Our calculations yield three qualitative conclusions about how long a forager should be willing to remain searching a patch without finding a prey. First, when travel time between patches is shorter, the forager can achieve a higher capture rate, and should stay a shorter time in each patch (see also Krebs et al., 1974). Second, for more variable environments (when a is small), the forager should stay a shorter time in each patch, since it becomes clear sooner whether a patch is poor. Third, the capture rate achieved is not very sensitive to the stopping time used. This robustness may hold quite generally for stochastic models (see also Green, 1984). Our most important conclusion does not involve the optimal time to stay in a patch under different conditions but rather the way the optimal capture rate depends
CENTRAL-PLACE
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on prey variability a m o n g patches. For given values of travel time between patches and to-and-from the nest, an optimal forager will achieve the highest capture rate when patch quality is least variable. As the variability in prey density increases a m o n g the patches the optimal capture rate decreases. To measure the importance of patch variability we might take a = 0.5 as a biologically realistic value for patch variability ( a = k in Southwood 1966) and see from Fig. 2(b) that the capture rate is only about one half to two thirds that achievable in the homogeneous patches assumed by Orians & Pearson. Differences of this magnitude must be important to the foragers. That is, the difference between whether patches are variable or identical is likely to be biologically important. While M c N a m a r a & Houston (1985) conclude, as we do, that the optimal time t* to remain in a patch decreases as patch variability increases, they also conclude that the optimal capture rate increases with patch variability. This is the opposite of our conclusion. This difference in conclusion is due to a difference in the way more variable and less variable environments are compared. To isolate the effect of patch variability we have c o m p a r e d environments with the same average rate of finding a prey in a patch. That is, we c o m p a r e environments with the same average prey density, but different degrees of variability a m o n g patches. M c N a m a r a & Houston (1985) isolate the effect of patch variability by comparing environments with the same average time between prey captures. They are not interested in prey density, but in the difficulty of finding a prey, or the difficulty of cracking a seed. If we were to follow their lead and compare environments by fixing the average time to find a prey for our model, then we would find that average prey density increases as patch variability increases. Thus, for our model, we interpret M c N a m a r a & Houston's finding that capture rate increases with patch variability as being due, not to the increase in patch variability, but rather to the increase in prey density. For example, in the most striking case of a change in the capture rate with increasing variability that M c N a m a r a & Houston (1985, p. 558, Table II) present, that rate increases from 0.6667 to 0.7639 as the parameter a changes from 0o (no variability) to 2. But, since they set/3 = a - 1 to keep the average time to find a prey constant, the average rate of finding prey ( = a//3), corresponding to average prey density, increases from 1 to 2 in this case. One should not be surprised if a doubling of prey density permits an increase in the capture rate. One should be surprised, instead, that the increase in capture rate is as low as the 15% that M c N a m a r a & Houston report. If, instead, average prey density is held constant (a//3 = 1), the capture rate would decrease from 0.6667 to 0.5000 as a changes from 0o to 2. M c N a m a r a & Houston (1985) suggest that their c o n c l u s i o n - - t h a t capture rate increases with patch variability--is what one might expect, since information is more valuable when patches are more variable. We disagree with the conclusion and with the intuitive explanation of it. They do not distinguish between the advantage of using information over not using information within a given environment and the advantage of being in an environment in which more information is available. We agree that in a more variable environment the advantage of using information over not using it is greater than it would be in a less variable environment. This is illustrated in our Fig. 1, in which capture rate is plotted against stopping
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time. It is seen that the advantage of using a shorter stopping time is greater when patches are more variable. This does not mean, however, that foragers using the optimal stopping time will achieve a higher rate in a more variable environment. This is seen by comparing the maximum capture rates shown in Fig. 1 for more and less variable environments. The reason that the availability of more information does not necessarily permit a higher capture rate is that obtaining information about a patch leads a forager to leave the patch about which the information has been obtained and go on to another patch about which the forager has no information. If no prey is found by a certain time, the evidence is that the patch is poor and should be left, while if a prey is found, the forager knows that the patch was good, but now must be left. In either case, the forager will next visit a patch about which it has no information. On the other hand, if patches can contain more than one prey, and if foragers can remain in good patches, or return to them, then foragers can use information about patch quality and achieve a higher capture rate in more variable environments. Ordinary foragers (not central-place foragers) can achieve their highest capture rate when patch quality is most variable (Green, 1980). Such foragers achieve their lowest capture rate when prey are randomly distributed (and patches do not vary in quality). Stewart-Oaten (1982) has noted this fact and suggested that natural selection operating on prey might favor mechanisms leading to random distribution. Since, for our model, the capture rate will be lowest when prey are most aggregated, the analogous suggestion is that for prey exploited by single-prey loaders, or predators that can be rendered single-prey loaders (perhaps by social behavior, including warning calls by the prey), selection should favor a clumped distribution. It happens that aggregation of prey lowers the predation rate and thus benefits the prey population as a whole, but it is not proper to argue that aggregation of prey is selected for this reason. However, in our model, individuals have a higher chance of avoiding predation as members of large aggregations. Thus, selection operating on individuals favors aggregation. Hamilton (1971) has shown how aggregation might be favored by selection on individuals, even if it increased the overall predation rate on the population. In our model, selection on individuals would favor aggregation, which, in turn, would lower the overall predation rate on the population. We assume that after returning from a visit to the nest, the forager resumes foraging in a new patch. If the forager remembers where it has found prey, and the prey have not been frightened away, then it should return to the patch where a prey was found, and our assumption does not hold. Thus, our model is likely to apply to foragers seeking mobile rather than immobile prey. We have presented our model in terms of central-place foraging, but the ideas also hold for any predator taking at most one prey per patch visit. An example is a hide-and-wait predator that warns or scares away other potential prey when it attacks. We thank John Krenz, Dave Schimpf and Allan Stewart-Oaten for reading and commenting on this paper. Most of the work was done during a visit to the Department of Zoology at Oxford, supported by the Graduate School of the University of Minnesota.
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REFERENCES ANDERSSON, M. (1978). ~eor. Pop. Biol. 13, 391. ANDERSSON, M. (1981). Ecology 62, 538. ARONSON, R. B. & GIVNISH, T. J. (1983). Ecology 64, 395. GREEN, R. F. (1980). Theor. Pop. Biol. 18, 244. GREEN, R. F. (1984). Am. Nat. 123, 30. HAMILTON, W. D. (1971). J. theor. Biol. 31, 295. KREBS, J. R., RYAN, J. & CHARNOV,E. L. (1974). Anita. Behav. 22, 953. LESSELLS, C. M. & STEPHENS, D. W. (1983). Anita. Behav. 31,238. MCNAMARA, J. & HOUSTON, A. (1985). Anita. Behav. 33, 553. ORIANS, G. H. & PEARSON, N. E. (1979). In: Advances in EcologicaISystems. (by Horn, D. J., Mitchell, R. D. & Stairs, G. R. eds). pp. 154-177. Columbus: Ohio State University Press. PIELOU, E. C. (1977). Mathematical Ecology. New York: Wiley Interscience. SOUTHWOOD, T. R. E. (1966). Ecological Methods. London: Chapman & Halt. STEWART-OATEN,A. (1982). Theor. Pop. Biol. 22, 410.
Central-place Foraging in a Patchy Environment RICHARD F. GREEN
Department of Mathematical Sciences, University of Minnesota, Duluth, Minnesota 55812, U.S.A. AND ADELINE TAYLOR NUNEZ
Forest Wildlife Project, Cloquet Forestry Center, 175 University Road, University of Minnesota, Cloquet, Minnesota 55720, U.S.A. (Received 12 June 1985, and in revised form 18 March 1986) Early models of central-place foraging treated animals that search for prey in identical, homogenous patches. If patches vary in quality, then optimal foraging requires strategies based on time spent in a patch, and not simply on the type or number of prey found. In particular, a forager that takes no more than one prey from a patch should leave a patch after searching unsuccessfully for a certain fixed time. When patches are more variable, the forager should stay a shorter time in each patch, and the resulting rate of delivering prey to the central place will be lower. This implies that aggregation should be favored by prey faced with a singleprey-loading predator.
1. Introduction Central-place foraging is a special case of foraging in which animals return with prey to a nest or larder, and carry a limited n u m b e r of prey on each return. Orians & Pearson's (1979) model of central-place foraging describes an animal foraging in h o m o g e n e o u s patches in which prey are found randomly at a known, constant rate. Orians & Pearson consider two different cases and treat a characteristic problem for each. (1) A single-prey loader takes one prey per visit to a patch. If prey vary in quality (the energy they yield to the forager, for example), then the forager should take the highest quality prey. The forager should be more selective when travel time t o - a n d - f r o m the nest is greater. (2) A multiple-prey loader may take several prey per visit to a patch. Such a forager must decide how many prey to take on each visit. The forager should take more prey when travel time to-and-from the nest is greater. Lessells & Stephens (1983) treat a special case o f Orians & Pearson's model for a single-prey loader, and state that it makes no sense for such an animal to leave a patch without finding a prey. This is true if patches are identical and homogeneous, as assumed by Orians & Pearson, but it is not true if patches vary in prey density. In this paper, we consider a model of a single-prey loader that forages in patches that vary in prey density. 35
0022-5193/86/210035+09 $03.00/0
O 1986 Academic Press Inc. (London) Ltd
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R.F. GREEN
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2. The Model
Imagine a central-place forager, perhaps a hawk or owl, which can carry only one prey item to the nest on each return, and which forages for prey in patches that vary in prey density. On each foraging excursion the animal leaves the nest, searches a patch for a time, and if no prey are found, moves on to another patch and searches it for a time. The animal moves from patch to patch until it captures a prey and returns to the nest. When the animal resumes foraging, it does not return to the patch where the last prey was captured. For simplicity, we assume that all prey are the same value. We assume that search within a patch is systematic, and that prey are encountered randomly, at rate r, which is proportional to the density of prey in the patch. The time to encounter a prey in a patch would have an exponential distribution with density
f(t)=rexp(-rt)
for t > 0
(I)
and the expected time until a capture would be ~z = 1/r. We assume that prey density varies from patch to patch, and that the encounter rate, r, is a random variable, having a gamma distribution with parameters ~ and/3. That is, r has probability density function
/3°
f(r)-F(a) r
e -~r
(2)
This mathematically convenient assumption implies that the number of prey encountered by a forager spending a fixed time in a patch has a negative binomial distribution (see Pieiou 1977, pp. 122-123). In our model, the average travel time between patches is t(1), and the average travel time back and forth from a patch to the nest is t(2), with t ( 2 ) > t(1). Our model has four parameters: a,/3, t(1) and t(2). Since patches vary in prey density, the longer a forager spends in a patch without finding a prey, the worse that patch is likely to be. The best strategy is to stay in each patch until a prey has been found, or until a fixed stopping time, t, is reached without finding a prey. Various choices of t could be made, but the best value, t*, maximizes the long-term average rate of obtaining prey (which we will refer to as the "capture rate"), given by
R = 1/(t(2)+ET),
(3)
where E T is the expected time spent foraging and traveling between patches per prey found. Notice that the value of t* depends on a,/3, and t(1), but not on t(2), the travel time to-and-from the nest, since the maximum capture rate is achieved by the rule that minimizes the expected time to find a prey while foraging. McNamara & Houston (1985) propose a model similar to ours for a slightly different problem. Considering a forager searching patches that contain exactly one prey, they assume that the time to find a prey is a random variable having an exponential distribution with mean v (corresponding to the difficulty of finding the prey), which is itself a random variable having a gamma distribution. The r in our model equals 1/v in McNamara & Houston's model. This rather subtle difference
CENTRAL-PLACE FORAGING
37
in the models leads to a substantial difference in conclusions when we consider the effect of environmental variability. CALCULATION
OF
ET
For each choice of a, 13 and t(1) we can find the optimal stopping time, t*, which minimizes ET. For any a, 13 and t, the probability that a forager will fail to find a prey after searching a patch for time t is given by Q(t) =
[/3/(/3 + t)]%
(4)
If T is the time that a forager would have to wait to find a prey in a randomly chosen patch, the probability distribution of T is given by P(T < - t) = F ( t ) --- 1 - Q ( t )
(5)
and its probability density is given by f ( t) = F'( t) = ct/3~[/3 + t] -~-~.
(6)
That is, the random variable T has a Pareto distribution with location parameter/3 and shape parameter c~. If a prey is found before time t in a patch, the expected time spent in the patch before finding that prey is
for a ¢ 1, and
for a = 1. The average time spent foraging on each excursion is given by E T = E [ t I T <- t] + ( t ( 1 ) + t ) Q / ( 1 - Q )
(8)
where ( t ( 1 ) + t) denotes the time spent foraging and traveling between patches for each patch in which a prey is not found, and Q / ( 1 - Q ) is the average number o f patches searched unsuccessfully. 3. Results
Without loss of generality, we assume that all environments have the same average prey density per patch, (for example, a / / 3 = 1, and, therefore, that a =/3). Thus, in analyzing our model, we need to consider two parameters, a and t(1), where a is a measure o f patch variability, and t(1) is related to the density of patches in the environment. We compare environments that have the same average prey density, but differ in (a) the variability in prey density among patches, and (b) the travel time between patches. We look at three things: (1) how the capture rate achieved depends on the strategy used (the time a forager is willing to remain in a patch
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without finding a prey) for ditterent parameters, (2) how the optimal value t* depends on the parameters, and (3) how the optimal capture rate depends on the parameters. For any set of parameter values we can find the capture rate, R, achieved by a forager staying in a patch for time, t. In Fig. 1, R is plotted against t for four different sets of parameter values. The rate does depend on t, but is rather insensitive to t, especially when patches are less variable. This insensitivity of the rate achieved to the strategy used is seen in other stochastic foraging models as well (Green, 1984). Notice that the capture rate is highest when the travel time between patches is short and variability of prey density among patches is low. Capture rate is lowest when the travel time between patches is long and variability of prey density among patches is high. 0-5( Manylow-variabilityporches 04( cc
~ F e w
~ 0 3( r~ o
Iow-voriobihtypatches
~ o t c h e s
o~ Few h~gh-vorlobJhtypotches
020 / /
8 010 000
V
000
i
r
i
I
100 200 3 O0 400 Stoppingtime,t
i
500
FIG. l. Long-term average capture rate, R, plotted against stopping time, t (the time a forager is willing to remain in a patch without finding a prey), for four different sets of environmental conditions. Travel time between patches, t( 1) = 0.2 ( m a n y patches), or t(1 ) = 1 (few patches), while patch variability, a = 0.5 (high variability), or a = 2 (low variability). In each case, average patch quality, a / f l = 1, and travel time to-and-from the nest, t(2) = 1.
The reason that a forager should leave a patch if no prey are found by time t* is that as time spent in the patch without finding a prey increases, the chance of finding a prey there decreases. When a patch is entered, prey are found at rate a/fl, but after time, t, has been spent in the patch without finding prey, the rate decreases to a/(/3 + t). Since we assume that a =/3 this rate is a / ( a + t). If a forager increases the time it is willing to stay in a patch before finding a prey, it will increase the chance of finding a prey in a patch (that is, it will decrease Q), and thus decrease the chance of having to waste time travelling between patches. On the other hand, being willing to stay in a patch for a longer time without finding prey will increase the time wasted searching poor patches. When travel time between patches, t(1), is longer, travel is more costly and the optimal value, t*, will be higher. When patches are more variable (that is, a is smaller), a forager discovers more quickly if a patch is poor, and t* will be lower. Figure 2(a) plots the optimal patch time, t*, against or, for two different values of t(1), the travel time between patches. As a increases, patches are less variable and a forager should be willing to stay longer in each patch. In Fig. 2(b), the
CENTRAL-PLACE 4 O0
39
FORAGING 0 50
(o)
Many patches t(1)=O 2
~320
040
Few patches t (1) = 1 0 / 1 o
~24c
8 "8 16C E
,t J
- 020 0
ooo o'4o o18o ~'zo ~o
2oo
i "~
///
Few patches f(1}=t 0
///
Many patches t(1)=O 2
0 8( 00(
//
8
// //
ts
050
/ I
010
/
/
/
/
/
0 O0
ooo o'4o o'8o ~'2o ~'~o 20o
Cl(lncreose as patch varlobi~ily decreases)
(:l (~ncremsesas patch voflabthty decreases)
FIG. 2. (a) Optimal stopping time, t*, and (b) long-term average capture rate achieved by a forager using the optimal stopping time, plotted aginst a, a measure of patch variability (increasing a means decreasing variability). Travel time between patches, t(1) = 0.2 (many patches), or t(1) = 1 (few patches). In each case, average patch quality, all3 = 1, and travel time to-and-from the nest, t(2) = 1.
maximum possible capture rate (that is, the rate achieved by the value of t* shown in Fig. 2(a)) is plotted against a. Notice that capture rate is highest when patches are least variable. If, for a//3 = 1 and t ( 2 ) = 1, patches are homogeneous (t~ =oo), the forager should stay in each patch until a prey is found (t* = ~ ) , and the capture rate would be R =0.5. This is the case considered by Orians & Pearson (1979). 4. Discussion REALISM A N D G E N E R A L I T Y OF THE M O D E L
The model we have considered is a special case of foraging, but the assumption that makes it different from other models--that patches vary in prey density--is realistic biologically. One of the consequences of our assumption is that foragers should sometimes leave a patch without finding a prey. This is consistent with observations of animals feeding. The particular model of patch variability we have assumed implies that the number of prey per patch of fixed size should have a negative binomial distribution. This distribution seems to fit some data for insects well (Southwood, 1966), and the choices o f a = 0-5 and 2 which we have used in Fig. 1 are biologically reasonable. Our assumption that all prey are the same leads to the conclusion that the optimal strategy, which consists of choosing a value of t*, does not depend on travel time to-and-from the nest. This conclusion will also hold if there are different types of prey of different value to the forager, but the forager does not choose among them. If several types of prey are considered, then the optimal choice of prey may depend on travel time, but, contrary to the conclusion o f Orians & Pearson (1979), who assumed that patches are homogeneous, the choice o f prey may also depend on the length o f time a forager has spent in a patch. If patches are homogeneous, then Orians & Pearson are correct that the decision whether to take a prey of given quality or to continue searching until a better prey is found does not depend on how long the forager has searched the patch, since the forager has learned nothing
40
R.F.
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from the length of his search. However, if patches vary in quality, then as more time is spent in a patch the forager should be willing to settle for poorer prey. If prey vary in value, V, to a forager, but not in the time necessary to capture and handle them, and if R is the highest possible capture rate, then a prey should be taken whenever found if V> Rt(2) (9) while a prey should never be taken if V< R[t(2) - t(1)].
(10)
Prey of intermediate value should be taken only if the forager has spent sufficient time in a patch without taking a better prey. The value of t(2), the travel time to-and-from the nest, is determined by how far the animal forages from the nest. Following Orians & Pearson (1979), we have treated t(2) as a constant. Of course, animals should prefer a short t(2), that is, they should forage near their nests. However, if animals forage in the vicinity of their nests, the area might become depleted of prey. Orians & Pearson (1979) suggest that the choice of where to forage involves a trade-off between the cost of travelling to distant patches and the cost of foraging in depleted nearby patches. Optimal foraging theory might contribute to an understanding of how animals use territories. There has been little work in this direction. Andersson (1978, 1981) has argued that efficient foraging would lead to a territory being used with intensity decreasing linearly away from the central place. The evidence for this idea has been disputed by Aronson & Givnish (1983), who suggest possible alternative patterns of territory use. These alternatives do not take advantage of the ideas of optimal foraging theory. We do not propose any solution to the problem of how animals use, or should use, their territories, but we do suggest that optimal foraging theory must be used, and that theory must take account of the prey distribution. For example, our results show that the optimal foraging strategy and the capture rate depend on prey distribution as well as average prey density. The choice of whether or not to search in partially depleted patches near the nest depends on the capture rate that can be achieved there, which, in turn, depends on the original prey distribution and the pattern of foraging which has depleted the patches. 5. Conclusions
Our calculations yield three qualitative conclusions about how long a forager should be willing to remain searching a patch without finding a prey. First, when travel time between patches is shorter, the forager can achieve a higher capture rate, and should stay a shorter time in each patch (see also Krebs et al., 1974). Second, for more variable environments (when a is small), the forager should stay a shorter time in each patch, since it becomes clear sooner whether a patch is poor. Third, the capture rate achieved is not very sensitive to the stopping time used. This robustness may hold quite generally for stochastic models (see also Green, 1984). Our most important conclusion does not involve the optimal time to stay in a patch under different conditions but rather the way the optimal capture rate depends
CENTRAL-PLACE
FORAGING
41
on prey variability a m o n g patches. For given values of travel time between patches and to-and-from the nest, an optimal forager will achieve the highest capture rate when patch quality is least variable. As the variability in prey density increases a m o n g the patches the optimal capture rate decreases. To measure the importance of patch variability we might take a = 0.5 as a biologically realistic value for patch variability ( a = k in Southwood 1966) and see from Fig. 2(b) that the capture rate is only about one half to two thirds that achievable in the homogeneous patches assumed by Orians & Pearson. Differences of this magnitude must be important to the foragers. That is, the difference between whether patches are variable or identical is likely to be biologically important. While M c N a m a r a & Houston (1985) conclude, as we do, that the optimal time t* to remain in a patch decreases as patch variability increases, they also conclude that the optimal capture rate increases with patch variability. This is the opposite of our conclusion. This difference in conclusion is due to a difference in the way more variable and less variable environments are compared. To isolate the effect of patch variability we have c o m p a r e d environments with the same average rate of finding a prey in a patch. That is, we c o m p a r e environments with the same average prey density, but different degrees of variability a m o n g patches. M c N a m a r a & Houston (1985) isolate the effect of patch variability by comparing environments with the same average time between prey captures. They are not interested in prey density, but in the difficulty of finding a prey, or the difficulty of cracking a seed. If we were to follow their lead and compare environments by fixing the average time to find a prey for our model, then we would find that average prey density increases as patch variability increases. Thus, for our model, we interpret M c N a m a r a & Houston's finding that capture rate increases with patch variability as being due, not to the increase in patch variability, but rather to the increase in prey density. For example, in the most striking case of a change in the capture rate with increasing variability that M c N a m a r a & Houston (1985, p. 558, Table II) present, that rate increases from 0.6667 to 0.7639 as the parameter a changes from 0o (no variability) to 2. But, since they set/3 = a - 1 to keep the average time to find a prey constant, the average rate of finding prey ( = a//3), corresponding to average prey density, increases from 1 to 2 in this case. One should not be surprised if a doubling of prey density permits an increase in the capture rate. One should be surprised, instead, that the increase in capture rate is as low as the 15% that M c N a m a r a & Houston report. If, instead, average prey density is held constant (a//3 = 1), the capture rate would decrease from 0.6667 to 0.5000 as a changes from 0o to 2. M c N a m a r a & Houston (1985) suggest that their c o n c l u s i o n - - t h a t capture rate increases with patch variability--is what one might expect, since information is more valuable when patches are more variable. We disagree with the conclusion and with the intuitive explanation of it. They do not distinguish between the advantage of using information over not using information within a given environment and the advantage of being in an environment in which more information is available. We agree that in a more variable environment the advantage of using information over not using it is greater than it would be in a less variable environment. This is illustrated in our Fig. 1, in which capture rate is plotted against stopping
42
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time. It is seen that the advantage of using a shorter stopping time is greater when patches are more variable. This does not mean, however, that foragers using the optimal stopping time will achieve a higher rate in a more variable environment. This is seen by comparing the maximum capture rates shown in Fig. 1 for more and less variable environments. The reason that the availability of more information does not necessarily permit a higher capture rate is that obtaining information about a patch leads a forager to leave the patch about which the information has been obtained and go on to another patch about which the forager has no information. If no prey is found by a certain time, the evidence is that the patch is poor and should be left, while if a prey is found, the forager knows that the patch was good, but now must be left. In either case, the forager will next visit a patch about which it has no information. On the other hand, if patches can contain more than one prey, and if foragers can remain in good patches, or return to them, then foragers can use information about patch quality and achieve a higher capture rate in more variable environments. Ordinary foragers (not central-place foragers) can achieve their highest capture rate when patch quality is most variable (Green, 1980). Such foragers achieve their lowest capture rate when prey are randomly distributed (and patches do not vary in quality). Stewart-Oaten (1982) has noted this fact and suggested that natural selection operating on prey might favor mechanisms leading to random distribution. Since, for our model, the capture rate will be lowest when prey are most aggregated, the analogous suggestion is that for prey exploited by single-prey loaders, or predators that can be rendered single-prey loaders (perhaps by social behavior, including warning calls by the prey), selection should favor a clumped distribution. It happens that aggregation of prey lowers the predation rate and thus benefits the prey population as a whole, but it is not proper to argue that aggregation of prey is selected for this reason. However, in our model, individuals have a higher chance of avoiding predation as members of large aggregations. Thus, selection operating on individuals favors aggregation. Hamilton (1971) has shown how aggregation might be favored by selection on individuals, even if it increased the overall predation rate on the population. In our model, selection on individuals would favor aggregation, which, in turn, would lower the overall predation rate on the population. We assume that after returning from a visit to the nest, the forager resumes foraging in a new patch. If the forager remembers where it has found prey, and the prey have not been frightened away, then it should return to the patch where a prey was found, and our assumption does not hold. Thus, our model is likely to apply to foragers seeking mobile rather than immobile prey. We have presented our model in terms of central-place foraging, but the ideas also hold for any predator taking at most one prey per patch visit. An example is a hide-and-wait predator that warns or scares away other potential prey when it attacks. We thank John Krenz, Dave Schimpf and Allan Stewart-Oaten for reading and commenting on this paper. Most of the work was done during a visit to the Department of Zoology at Oxford, supported by the Graduate School of the University of Minnesota.
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